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What is an exponential growth model?

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An exponential growth model describes a process where the quantity increases at a rate proportional to its current value, often represented by the formula: P(t) = P_0 * e^(rt), where P_0 is the initial amount, r is the growth rate, and t is time.

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How do you calculate the future value of an investment with compound interest?

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The future value (FV) can be calculated using the formula: FV = P(1 + r/n)^(nt), where P is the principal amount, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.

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What does it mean for a population to increase by a certain percentage each year?

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It means that each year, the population grows by that percentage of its current size. For example, a 15% increase means the population grows by 15% of its current number every year.

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What is the formula for exponential growth in terms of time?

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The formula is: N(t) = N_0 * (1 + r)^t, where N(t) is the population at time t, N_0 is the initial population, r is the growth rate (as a decimal), and t is the time in years.

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How do you model the growth of a quantity that increases by a fixed percentage each time period?

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You can model it using the formula: A = P(1 + r)^t, where A is the amount after time t, P is the initial amount, r is the growth rate, and t is the number of time periods.

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What is the significance of the base 'e' in exponential growth?

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The base 'e' (approximately 2.718) is used in continuous growth models, representing the natural growth rate of processes that grow continuously rather than at discrete intervals.

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How do you determine the growth rate from an exponential growth problem?

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The growth rate can be determined by rearranging the exponential growth formula to solve for r, often using logarithms to isolate r.

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What is the difference between linear and exponential growth?

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Linear growth increases by a constant amount over time, while exponential growth increases by a percentage of the current amount, leading to faster growth as the quantity increases.

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How can you apply exponential growth models to real-world scenarios?

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Exponential growth models can be applied to populations, investments, and any scenario where growth accelerates over time, such as technology adoption or viral spread.

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What is the formula for calculating the value of an investment after a certain number of years with compound interest?

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The formula is: A = P(1 + r/n)^(nt), where A is the amount of money accumulated after n years, including interest, P is the principal amount, r is the annual interest rate, n is the number of times that interest is compounded per year, and t is the number of years.